Achieved power for Welch’s t-test

When to use it

Assess the sensitivity of a comparison of two independent means when the groups are expected to differ in spread, or in size and spread together.

Not for

With equal spreads and equal groups, Welch’s and Student’s tests have almost the same power; use the two-sample tab.

Notation

$n_1, n_2$: Group sizes, n₂ = ⌈r n₁⌉.
$\Delta$: Difference between the group means.
$\sigma_1, \sigma_2$: Standard deviation in each group.
$\nu$: Welch–Satterthwaite degrees of freedom.

Formulas

Unpooled variance and degrees of freedom

v=\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}, \quad \nu=\frac{v^2}{\frac{(\sigma_1^2/n_1)^2}{n_1-1}+\frac{(\sigma_2^2/n_2)^2}{n_2-1}}, \quad \lambda=\frac{|\Delta|}{\sqrt v}

The spreads are not pooled, and buy their own degrees of freedom [1, 2].

Power from the rejection region

\text{Power}=P(T_{\nu,\lambda}>t_{1-\alpha/\text{tails},\nu}) + I_{\text{tails}=2}P(T_{\nu,\lambda}<-t_{1-\alpha/\text{tails},\nu})

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Parameters

Group 1 size $n_1$

Values: Integer, at least 2

Participants available in group 1; group 2 follows from the allocation ratio.

Allocation ratio $r$ Optional

Values: Positive; 1 for equal groups

Group 2 has n₂ = ⌈r n₁⌉ participants.

Mean difference $\Delta$

Values: Any non-zero value in outcome units

The expected difference between the group means.

SD in group 1 $\sigma_1$

Values: Positive

Expected standard deviation in group 1.

SD in group 2 $\sigma_2$

Values: Positive

Expected standard deviation in group 2.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

The groups are independent and the outcome is approximately normal within each; the spreads may differ.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 60 in group 1 and 120 in group 2, a mean difference of 5, and SDs of 10 and 15 at α = 0.05: v = 10²/60 + 15²/120 = 3.54, so λ = 5/√3.54 = 2.66.

Needed for analysis: 60 · To approach: 180

Open this example in the calculator

Validation

At the size the Welch calculator returns, this power reaches that calculator’s target, and one fewer falls short; it is checked against an independent calculation in R.

How to cite

StatWisely. Achieved power for Welch’s t-test. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-welch/

Cite the original references for the formulas your design uses as well.

References

1.Welch BL. The generalization of “Student’s” problem when several different population variances are involved. Biometrika. 1947;34(1–2):28–35.
2.Satterthwaite FE. An approximate distribution of estimates of variance components. Biometrics Bulletin. 1946;2(6):110–114.